Page 81 - ICSE Math 8

Properties of complement of a set
(a) Union of a set and its complement is equal to the universal set, i.e., A ∪ A′ = U.
(b) Intersection of a set and its complement is the empty set, i.e., A ∩ A′ = φ.
(c) The double complement of a set is the set itself, i.e., (A′)′ = A.
(d) The complement of an empty set is a universal set, i.e., φ′ = U – φ = U.
(e) The complement of a universal set is an empty set, i.e., U′ = U – U = φ.
(f) De Morgan’s Laws
(i) The complement of union of two sets is equal to the intersection of their complements, i.e.,
(A ∪ B)′ = A′ ∩ B′
(ii) The complement of intersection of two sets is equal to the union of their complements, i.e.,
(A ∩ B)′ = A′ ∪ B′
Example 15: If U = {x : x is an integer, –5 ≤ x < 8}, A = {–2, 0, 3, 4}, B = {–4, –1, 2, 4, 6} and
C = {–1, 0, 1, 2, 3}, then find the following sets.
(a) A′ (b) (A ∪ B)′ (c) A′ ∩ B′
(d) B′ ∪ C′ (e) A ∩ C′
Solution: The universal set in roster form is given by U = {–5, –4, –3, –2, –1, 0, 1, 2, 3, 4, 5, 6, 7}.
(a) A′ = U – A = {–5, –4, –3, –1, 1, 2, 5, 6, 7}
(b) A ∪ B = {–4, –2, –1, 0, 2, 3, 4, 6}
\ (A ∪ B)′ = U – (A ∪ B) = {–5, –3, 1, 5, 7}
(c) A′ = {–5, –4, –3, –1, 1, 2, 5, 6, 7}, B′ = {–5, –3, –2, 0, 1, 3, 5, 7}
\ A′ ∩ B′ = {–5, –3, 1, 5, 7}
(d) B′ = {–5, –3, –2, 0, 1, 3, 5, 7}, C′ = {–5, –4, –3, –2, 4, 5, 6, 7}
\ B′ ∪ C′ = {–5, –4, –3, –2, 0, 1, 3, 4, 5, 6, 7}
(e) C′ = {–5, –4, –3, –2, 4, 5, 6, 7}
\ A ∩ C′ = {–2, 4}

EXERCISE 6.3

1. Find the union and intersection of each of the following pairs of sets.
(a) A = {3, 6, 9}, B = {2, 3, 5, 7}
(b) C = {x : x is a natural number and multiple of 2, x ≤ 18},
D = {x : x is a natural number < 10}
(c) E = {p, q, r, s, t}, F = {p, s, t, v}
2. If A and B are two sets such that A ⊂ B, then what is A ∩ B?
3. If A = {6, 7, 8, 9, 10}, B = {2, 4, 6, 8} and C = {3, 6, 9, 12}, prove that:
(a) A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C) (b) A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C)
4. If X = {x : x is an even natural number}, Y = {x : x is an odd natural number},
Z = {x : x is a natural number} and V = {x : x is a prime number}, find the following sets.

(a) X ∩ Y (b) Y ∩ V (c) Y ∩ Z (d) X ∩ V (e) Z ∩ V
5. Which of the following pairs are disjoint?
(a) {3, 4, 5, 6, 7} and {x : x ∈ N, 7 ≤ x ≤ 10}
(b) {u, v, w, x, y, z} and {p, q, r, s, t}
(c) {x : x ∈ Z, –3 < x ≤ 1} and {–1, 0, 1, 2}
(d) {x : x is an even natural number} and {x : x is an odd natural number}

69